Sum Of Series Research Articles

Sum of series and new relations for Mittag-Leffler functions

Sum of series and new relations for Mittag-Leffler functions

Norm and Numerical Radius Inequalities for Sums of Power Series of Operators in Hilbert Spaces

The main focus of this paper is on establishing inequalities for the norm and numerical radius of various operators applied to a power series with the complex coefficients h(λ)=∑k=0∞akλk and its modified version ha(λ)=∑k=0∞|ak|λk. The convergence of h(λ) is assumed on the open disk D(0,R), where R is the radius of convergence. Additionally, we explore some operator inequalities related to these concepts. The findings contribute to our understanding of operator behavior in bounded operator spaces and offer insights into norm and numerical radius inequalities.

A note on Laplace Adomian decomposition method on a linear problem

In last two decades, Laplace Adomian Decomposition Method (LADM) is vastly used to solve non-linear (or even fractional order) differential equations. The method approaches the solution with the partial sums of function series. However, it is not easy to show that the limit of the function series is the exact solution of the problem. In this article, we consider a simple problem such as homogenous second-order linear ordinary differential equation with constant coefficients. We prove analytically that the LADM gives the right exact solution to the considered problem.

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in Cs for some positive integer s>1. We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials.

Asymptotic estimates for deviations of Fejér means on Poisson integrals

The work is devoted to the investigation of problem of approximation of continuous periodic functions of a real variable by trigonometric polynomials generated by linear methods of summation of Fourier series. The simplest example of the process of linear approximation of periodic functions is approximation of functions by partial sums of their Fourier series. However, sequences of partial Fourier sums are not uniformly convergent on the whole class of continuous periodic functions. Therefore, numerous studies are devoted to the study of approximation properties of approximating methods, which are generated by different transformations of sequences of partial sums of the Fourier series and allow obtaining sequences of trigonometric polynomials that are uniformly convergent for the entire class of continuous functions. In particular, Fejér means have been intensively studied in recent decades. One of the important tasks in this direction is study of the asymptotic behavior of the upper bounds of the deviations of trigonometric polynomials for a fixed class of periodic functions. The aim of the work is to systematize the known results concerning the approximation of classes of periodic functions of high smoothness by arithmetic means of Fourier sums, and to present new facts obtained for a more general case. In paper we investigate the asymptotic behavior of upper bounds on classes of Poisson integrals of periodic functions of real variable of deviations of linear means of Fourier series, which are constructed using the Fejér summation method. We study the classes consist of analytic functions of a real variable, which can be regularly extended into the corresponding strip of the complex plane. In the work, asymptotic inequalities for the upper bounds of the deviations of the Fejér means on the class of Poisson integrals were found. In certain case of parameters defining the class, the obtained formulas coincide with previously known asymptotic equalities.

ON THE APPROXIMATION OF UNIFORM ALMOST PERIODIC FUNCTIONS BY CERTAIN SUMS AND INTEGRALS

The paper investigates some issues of approximation of almost-periodic Bohr functions from partial sums of the Fourier series and the Marcinkevich averages, when the Fourier exponents (the spectrum of the function) of the functions under consideration have a limit point at infinity. The question of the deviation of a given function f(x) from its partial sums of the Fourier series is investigated, depending on the rate of tendency to zero of the magnitude of the best approximation by a trigonometric polynomial of bounded degree. Here, when determining the Fourier coefficients, instead of the function under consideration, some arbitrary, real, continuous function , is taken, which in a given interval is equal to one, and in other cases is zero. Then, similarly, the upper estimate of the deviation of the almost-periodic in the sense of the Bohr function by the Marcinkevich averages is established.

Expressing Solutions of Bessel Differential Equation in terms of Hypergeometric Functions

The BDE (Bessel differential equation) is a second-order linear ordinary differential equation (ODE), and it is considered one of the most significant differential equations because of its extensive applications. The solutions of this differential equation are called Bessel functions. These solutions can be expressed in terms of hypergeometric functions, and in this study, the Bessel functions of the first kind are worked. Hypergeometric functions are the sum of a hypergeometric series. A series ∑▒u_n is known as hypergeometric when the ratio u_(n+1)⁄u_n is a rational function of n. BDE has coefficients with variables; therefore, it is solved by the Frobenius approach. We implement some mathematical steps based on the definition of hypergeometric functions to express the solutions in terms of hypergeometric function. The result shows that the solution of the BDE in terms of hypergeometric function is f(x)=a_0 x^p (_1^ )F_2 (1;1+(p+ρ)/2,1+(p-ρ)/2;x^2⁄4) and the two linearly independent solutions for the BDE of order ρ in terms of hypergeometric function are 〖f_1 (x)=a〗_0 x^ρ (_0^ )F_1 (1+ρ;x^2⁄4) and〖〖 f〗_2 (x)=a〗_0 x^ρ (_0^ )F_1 (1-ρ;x^2⁄4).

On orthogonal Laurent polynomials related to the partial sums of power series

Let \(f(z) = \sum_{k=0}^\infty d_k z^k\), \(d_k\in\mathbb{C}\backslash\{ 0 \}\), \(d_0=1\), be a power series with a non-zero radius of convergence \(\rho\): \(0 <\rho \leq +\infty\). Denote by \(f_n(z)\) the \(n\)-th partial sum of \(f\), and \(R_{2n}(z) = \frac{ f_{2n}(z) }{ z^n }\), \(R_{2n+1}(z) = \frac{ f_{2n+1}(z) }{ z^{n+1} }\), \(n=0,1,2,\dots\). By the general result of Hendriksen and Van Rossum there exists a unique linear functional \(\mathbf{L}\) on Laurent polynomials, such that \(\mathbf{L}(R_n R_m) = 0\), when \(n\not= m\), while \(\mathbf{L}(R_n^2)\not= 0\), and \(\mathbf{L}(1)=1\). We present an explicit integral representation for \(\mathbf{L}\) in the above case of the partial sums. We use methods from the theory of generating functions. The case of finite systems of such Laurent polynomials is studied as well.

Об интегральном подходе при использовании метода коллокации

The article describes a matrix method of polynomial Chebyshev approximation using an integral approach to construct a solution to a nonhomogeneous fourth-order differential equation with mixed boundary conditions of the first kind. The proposed method is based on the expansion of the fourth-order derivative of the desired function into a series in terms of Chebyshev polynomials of the first kind and the representation of the partial sum of this series as a product of matrices whose elements are, respectively, the Chebyshev polynomials and the coefficients in this expansion. In this paper, using analytical formulas for calculating integrals of Chebyshev polynomials, we obtain a representation of the desired function in terms of the product of the matrices defined above. The use of points of extrema and zeros of Chebyshev polynomials of the first kind as nodes, as well as the properties of the sums of products of Chebyshev polynomials at these points, made it possible to reduce the boundary value problem by the collocation method to a system of inhomogeneous linear algebraic equations with a sparse matrix of this system. It is shown that the solution constructed in this way satisfies the differential equation at all nodes, including the boundary ones, in contrast to the approximate solution obtained by approximating the exact solution in the form of a finite sum of the Chebyshev series. The effectiveness of the proposed method is demonstrated by considering a boundary value problem with a known analytical solution. The convergence analysis of the constructed solution is carried out.

On The Generalized Leonardo Quaternions and Associated Spinors

In this paper, we introduce and study a new family of sequences called the generalized Leonardo spinors by defining a linear correspondence between the generalized Leonardo quaternions and spinors. We start with defining the generalized Leonardo quaternions and then present their some important properties such as Binet type formula, Catalan’s identity, d’Ocagne’s identity, series sums, etc. We give some interrelations of these quaternions with the Fibonacci and Lucas quaternions. Then, we present the generating functions, sum formulae, various well-known identities, etc. for the Leonardo spinors and show their connection with the Fibonacci and Lucas spinors.

NON-STATIONARY WAVES IN A FUNCTIONALLY GRADED VISCOELASTIC PLANE-PARALLEL LAYER2

The propagation of nonstationary disturbances in a plane-parallel layer of infinite extent, which arise as a result of normal dynamic loading of one of its surfaces when the other surface is stationary, is investigated. The material of the layer is assumed to be functionally graded and, at the same time, viscoelastic. The hereditary properties of such a material are taken into account using linear integral Boltzmann–Volterra relations with regular kernels in the form of partial sums of the Prony's series. The peculiarity of this work is that here the parameters of the relaxation kernels are considered continuous functions of the transverse coordinate in the same way as other physical and mechanical characteristics. A method was used for the study, which consists in replacing a functionally graded material with an approximating layered homogeneous structure with continuity conditions at the contact of homogeneous layers. The solution of the nonstationary dynamic linear viscoelasticity problem for a package of plane homogeneous viscoelastic layers is presented in a special form, which greatly simplifies its numerical implementation, especially with a large number of layers with different hereditary properties. This made it possible to successfully apply this method and carry out a series of calculations using an efficient algorithm. Transient wave processes are investigated in the case when the parameters of a functionally graded material are non-monotonic functions of the transverse coordinate, symmetrical with respect to the middle surface of the layer. A comparison of transient wave processes for different types of these functions is carried out. The convergence of the calculation results with an increase in the number of approximating homogeneous layers with a continuous dependence of the external load on time is confirmed. The significant influence of both heterogeneity and viscosity of the material on nonstationary wave processes has been established.

ИССЛЕДОВАНИЕ РЕЗОНАНСНЫХ ЯВЛЕНИЙ КРУТИЛЬНЫХ КОЛЕБАНИЙ ВАЛОПРОВОДОВ ГРЕБНЫХ ВИНТОВ

The problem of torsional vibrations of propeller shaft lines is considered in the work. A two-stage rod model with distributed parameters has been adopted for the study of dynamic processes in contrast to the traditional method based on discrete masses. The active load on the engine side, as in the classical approach, is approximated by the partial sum of the Fourier series. The comparison of the calculation results according to the proposed model with the results known in the literature has been performed. It is established that both calculation methods give excessive values of forces and deformations in resonance with harmonics, for which the phase component is a multiple of the engine cycle.

APPROXIMATION OF FUNCTIONS BY THE MATRIX MEANS IN WEIGHTED ORLICZ SPACES

Abstract. In this work, the approximation properties of the matrix submethods in weighted Orlicz spaces L T M , are investigated. We obtain some results related to trigonometric approximation using matrix submethods of partial sums of Fourier series of functions in weighted Orlicz spaces.The degree of trigonometric approximations by the matrix methods to the functions have been investigated in weighted Orlicz spaces. The error of estimations in this work is obtained in more general terms.

Mathematical modeling of bending of a thin orthotropic plate clamped along the contour

Within the framework of Kirchhoff’s theory, a new approach to constructing a solution to the problem of modeling the bending of a thin rectangular orthotropic plate clamped along the contour, which is under the influence of a load normally distributed over its surface, is proposed. the solution to the inhomogeneous biharmonic equation for an orthotropic plate is obtained in the form of a partial sum of a double series in Chebyshev polynomials of the first kind. To find the coefficients in this expansion, the boundary value problem is reduced by the collocation method to a system of linear algebraic equations in matrix form using the properties of these polynomials. Based on matrix and differential transformations, expressions for bending moments and shearing forces are obtained. the results of calculations of the bending of the middle surface of the plate under different loads on the plate are presented, which demonstrate the effectiveness of the proposed approach.

ЕКВІВАВАЛЕНТНІСТЬ ФУНДАМЕНТАЛЬНОГО СПЛАЙНУ ТА ФУНКЦІЇ ГРІНА ДЛЯ ПОБУДОВИ ТОЧНОГО СКІНЧЕННОВИМІРНОГО АНАЛОГА КРАЙОВОЇ ЗАДАЧІ ДЛЯ ЗВИЧАЙНОГО ДИФЕРЕНЦІАЛЬНОГО РІВНЯННЯ 4-ГО ПОРЯДКУ

The author considers a problem with the main and natural boundary conditions on an interval. A new method for constructing an exact discrete analog of the problem is proposed. The method deals with the projection of the differential equation on local splines, formed by the fundamental system of solutions of the Cauchy problems for the homogeneous equation. A system of linear algebraic equations with a 5-diagonal matrix is obtained for the values of the exact solutions of the original problem at the points of a uniform grid. To implement an exact analog, we recommend using high-order accuracy schemes which are formed by partial sums of series in even powers of the grid step for solving the Cauchy problems. Keywords: boundary-value problem, ordinary differential equation of the 4th order, Cauchy problem, Wronskian, local spline, superposition of solutions, exact discrete analog, system of linear algebraic equations, 5-diagonal matrix.

Approximate averaged bounded synthesis for a parabolic process with two switching points

In this paper we analyze the problem of the optimal bounded control with semi-definite quality criterion for parabolic process with rapidly oscillating coefficients on finite time interval. The case when optimal control goes out both on lower, and on upper restriction, that is has two switch-points is considered. Our aim is to build a synthesized control which is practically preferable to program one. But after Fourier method application such a control will contain coefficients which are expressed by series and, in addition, it irregularly depends on small parameter ε. Thus, for practical application it is natural to restrict infinite series by finite sums and to get rid of dependence on a small parameter using average procedure. So, besides the law of optimal synthesis also approximate averaged feedback control, which provides control system behavior that is close to optimal one and thus has a series of advantages from the practical application point of view, is proposed and proved. The efficiency of approximate averaged control construction procedure consisted of cutting to finite sums of series in optimal control and replacing of Fourier coefficients nonregular depended on small parameter by corresponding average values is illustrated by an example of controlled system for parabolic process. We compare the properties of the averaged control and the sequence of optimal controls calculated at the different values of the small parameter. For comparison, we use the switching points of optimal control and averaged control, deviation between optimal control and averaged control, the difference in the values of the quality criterion on optimal control and averaged control. In considered example the precision of quality criteria value on approximate control has ε-order and for sufficiently small ε the precision is one order better then ε.

Comparative analysis of the approximation of a probabilistic indicator of functional stability using Bernstein polynomials and feed­forward neural networks

The use of multi-machine information systems is becoming an increasingly popular component of a wide variety of fields of activity. The increase in the number of problems solved through information systems and the increase in their complexity makes the study of the functional stability of the systems under consideration increasingly relevant. The functional stability of an information system is understood as the ability of this system to perform specified functions under the influence of negative influences. At the moment, a number of indicators have been developed to evaluate functional stability numerically. One such indicator is the convolution of the connectivity matrix. Convolution of the connectivity matrix, despite its completeness, has one very significant drawback: its calculation is a rather complicated procedure. Based on this, the question of approximate calculation of this indicator is logical. Since the convolution of the connectivity matrix can be considered as a function of the probability of serviceability of communication lines, an attempt to use certain methods of approximation theory is obvious. At the moment, methods of approximation theory are quite developed. Some of these methods are quite well researched, while others are just gaining popularity. The first group includes Lagrange interpolation polynomials, Legendre polynomials, Bernstein polynomials, splines, partial sums of series, etc., and the second group includes machine learning models, including feedforward neural networks, regression models, and others. Accordingly, the question immediately arises: which of these methods would allow better approximation of the convolution of the connectivity matrix and under what conditions? This paper provides a comparative analysis of the quality of approximation of the functional connectivity indicator of an information system based on the convolution of the connectivity matrix of this system using often considered feed-forward neural networks and using Bernstein polynomials, which, unfortunately, are not often considered despite their remarkable properties. The specifics of using each of these methods when approaching are also demonstrated and, based on this, it is indicated when which of these methods is better to use.

A Lower Bound in a Law of the Iterated Logarithm for Sums of Symmetric and Independent Random Variables

The law of the iterated logarithm for tail sum, abbreviated Tail LIL, was first introduced by R.Salem and S. Zygmund for sums of lacunary series. Tow and Teicher later introduced a corresponding result for independent random variables. Our article takes a different approach and focuses on obtaining one sided Tail LIL for the sums of independent and identically distributed symmetric random variables.

A simple yet efficient solution for TE scattering from a right trapezoidal groove

AbstractA simple yet efficient solution to the problem of transverse electric (TE) scattering by a two‐dimensional right trapezoidal groove placed in a perfect electric conductor (PEC) surface is presented. Using superposition of plane waves and applying the Neumann boundary condition at the groove walls, the transverse magnetic field is expressed as the sums of infinite series of waveguide modes. Considering the equivalent magnetic current on the aperture, a Fredlholm's integral equation with logarithmic singular kernel is extracted and then discretised by a collocation method based on a piecewise constant approximant. The extracted linear system of equations can be easily solved by the matrix methods for the unknown coefficients. Numerous examples are presented to verify the results and show the ability of the proposed method. Finally, this method is employed to examine the effect of the groove parameters and incidence angle on the scattering signatures.

An asymptotic development of the Poisson integral for Laguerre polynomial expansions

The purpose of this paper is the study of the rate of convergence of Poisson integrals for Laguerre expansions. The convergence of partial sums of Fourier series of functions in Lp spaces was studied, for several classes of orthogonal polynomials. In the Laguerre case Askey and Waigner proved convergence for functions f∈Lp0,+∞ with 4/3<p<4. In this paper we deal with the Poisson integral Arf0<r<1 which arises by applying Abel’s summation method to the Laguerre expansion of the function f. About 50 years ago, Muckenhoupt intensively studied the Poisson integral for the Laguerre and Hermite polynomials. Among other things he proved pointwise convergence, the convergence by norm, and that the Poisson integral is a contraction mapping in Lp0,∞. Toczek and Wachnicki gave a Voronovskaja-type theorem by calculating the limit 1−r−1Arfx−fx as r→1−, provided that f′′x exists. We generalize this formula by deriving a complete asymptotic development. All its coefficients are explicitly given in a concise form. As an application we apply extrapolation methods in order to improve the rate of convergence of Arfx as r→1−.